docs: restructure knowledge base into tree layout + gA extraction module

- Organize into tree structure: corrections/, nuclear-structure/, gA-extraction/, refs/ - Move all correction notes to corrections/ (01-09) - Move nuclear structure to nuclear-structure/ with index - Add gA-extraction/ with SSM algorithm, sNME/ESSM formalism, Tc99 case study, and implementation TODO - Reorganize refs by topic (electromagnetic/, forbidden-decay/) - Update all wikilinks to new paths - Remove radiative_spec.md (code algorithm, not reference) - Lint: all wikilinks verified valid - Comprehensive gA extraction research based on Paulsen et al. (2024) and Song et al. (2025)
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+# First-forbidden transitions in the reactor anomaly
+
+**Authors:** L. Hayen, J. Kostensalo, N. Severijns, and J. Suhonen
+**Affiliations:**
+
+1. Instituut voor Kern- en Stralingsfysica, KU Leuven, Belgium
+2. Department of Physics, University of Jyväskylä, Finland
+**Date:** August 23, 2019
+**Source:** arXiv:1908.08302v1 [nucl-th]
+
+---
+
+## Abstract
+
+This paper describes microscopic calculations of dominant forbidden transitions in reactor antineutrino spectra above 4 MeV using the nuclear shell model. By incorporating Coulomb corrections in the most complete way, the authors calculate the shape factor with high fidelity, showing strong deviations from allowed approximations. Despite small differences in ab initio electron cumulative spectra, large differences (several percent) are found in antineutrino spectra. The authors propose a parametrization of forbidden spectra and derive spectral corrections and uncertainties using Monte Carlo techniques. They conclude that a correct treatment of forbidden transitions is indispensable for solving both the reactor normalization anomaly and the spectral shoulder.
+
+## I. Introduction
+
+The field of short baseline reactor neutrinos has seen significant activity, driven by long-standing issues like the LSND anomaly [1, 2], GALLIX & SAGE collaborations [3], and the Reactor Antineutrino Anomaly (RAA) [4, 5]. The RAA involves a normalization deficit and a "5 MeV bump" in the spectrum [8–12]. Theoretical predictions often rely on the Huber and Mueller et al. treatments [14, 15], which introduced strong approximations for forbidden transitions. This work aims to provide a more complete analysis using the nuclear shell model for transitions above 4 MeV.
+
+## II. $\beta$ Decay Formalism
+
+The treatment of forbidden $\beta$ decays is complex, involving kinematic, nuclear, and Coulomb terms. The transition matrix element is given by:
+
+$$
+\mathcal{M}_{fi} = \int \mathrm{d}^3r \, \bar{\phi}_e(\vec{r}, \vec{p}_e) \gamma^\mu (1+\gamma^5) v(\vec{p}_\nu) \times \int \frac{\mathrm{d}^3s}{(2\pi)^3} e^{i\vec{s}\cdot\vec{r}} \frac{1}{2} [\langle f | \hat{p}(f) + \vec{p}_e - \vec{s} | V_\mu + A_\mu | i \rangle \langle \vec{p}_i \rangle + \langle f | \hat{p}(f) | V_\mu + A_\mu | i \rangle \langle \vec{p}_e + \vec{s} \rangle]
+$$
+
+The $\beta$ spectrum shape is traditionally written as:
+
+$$
+\frac{\mathrm{d}N}{\mathrm{d}W} = \frac{G_V^2 V_{ud}^2}{2\pi^3} p W (W-W_0)^2 \times F(Z,W) C(Z,W) K(Z,W)
+$$
+
+where $C(Z,W)$ is the shape factor. The shape factor depends on:
+
+1. Spin-change and kinematic factors.
+2. Finite size corrections proportional to $R^n$.
+3. Coulomb corrections proportional to $(\alpha Z)^n$.
+
+### A. Nuclear structure
+
+Forbidden transitions correspond to $\beta$ decays where Fermi and Gamow-Teller matrix elements are zero. The authors use the Behrens-Bühring formalism [23] to label nuclear structure form factors using quantum numbers $K, L, s$. For first-forbidden $\beta$ transitions ($\Delta J = 0, 1, 2$ and parity change $\pi_i \pi_f = -1$), up to 6 form factors contribute.
+
+### B. Coulomb corrections
+
+Coulomb corrections are split into static Coulomb renormalization and convolution distortion. The authors note that neglecting electron mass and Coulomb interaction leads to a symmetric shape factor, an argument often used to neglect forbidden transitions in the RAA context, which they argue is invalid.
+
+### C. Breakdown of usual approximations
+
+The $\xi$ approximation, often used to simplify shape factors, is shown to be questionable for relevant fission fragments where $\alpha Z / 2R \sim W_0$.
+
+## III. Data Selection & Handling
+
+The study relies on the ENDF/B-VIII.0 decay data library [38] and ENSDF data. To address the "Pandemonium effect" [37], where high-energy $\gamma$ decays are missed, the authors use TAGS (Total Absorption Gamma Spectroscopy) data [34–36] to correct branching ratios.
+
+## IV. Shape Factor Calculation
+
+The authors calculated shape factors for 36 dominant forbidden transitions using the NUSHELLX@MSU shell model code [48].
+
+**Table I: Dominant forbidden transitions above 4 MeV**
+*(Selected entries from Table I)*
+
+| Nuclide | $Q_\beta$ (MeV) | $E_{ex}$ (MeV) | BR (%) | $J_i^\pi \to J_f^\pi$ | FY (%) |
+| :--- | :--- | :--- | :--- | :--- | :--- |
+| $^{89}$Br | 8.3 | 0 | 16 | $3/2^- \to 3/2^+$ | 1.1 |
+| $^{90}$Rb | 6.6 | 0 | 33 | $0^- \to 0^+$ | 4.5 |
+| $^{91}$Kr | 6.8 | 0.11 | 18 | $5/2^+ \to 5/2^-$ | 3.5 |
+| $^{92}$Rb | 8.1 | 0 | 95.2 | $0^- \to 0^+$ | 4.8 |
+| ... | ... | ... | ... | ... | ... |
+
+The calculated shape factors (Fig. 3) show significant deviations from unity, especially for pseudovector transitions.
+
+## V. Spectral Changes
+
+Comparisons with existing literature (Fang and Brown [17]) show good agreement for $^{140}$Xe but discrepancies for $^{136}$Te, highlighting the sensitivity to nuclear structure details. The inclusion of forbidden shape factors leads to a downward trend in the electron spectrum below 4 MeV and an increase at higher energies (Fig. 5).
+
+## VI. Improved Forbidden Transition Treatment
+
+To generalize results to the entire database, the authors propose a parametrization of the shape factor:
+
+$$
+C = 1 + aW + b/W + cW^2
+$$
+
+They use Gaussian kernel density estimation to determine the distributions of fit parameters $a, b, c$ (Fig. 8, 9) and employ Monte Carlo techniques to estimate uncertainties.
+
+## VII. Updated Summation Calculations
+
+The authors combine their results with the ILL data set [47]. They compare three approaches:
+
+1. Treating transitions as allowed with weak magnetism correction.
+2. Treating them as allowed with $C=1$.
+3. Using the calculated forbidden shape factors.
+
+**Figure 14:** Relative change to cumulative antineutrino spectra. A bump appears between 4 and 7 MeV with a magnitude of up to 4.5%.
+
+**Table III:** Difference in integral and IBD flux compared to Huber-Mueller results.
+
+| | $^{235}$U | $^{238}$U | $^{239}$Pu | $^{241}$Pu |
+| :--- | :--- | :--- | :--- | :--- |
+| $\phi$ | 0.2(2) | 0.4(5) | 0.2(2) | 0.3(2) |
+| $R_{IBD}$ | 0.8(5) | 2.3(10) | 0.7(5) | 0.7(6) |
+
+## VIII. Reactor Spectrum Changes
+
+The study investigates the spectral shoulder (Fig. 17) and rate anomaly (Fig. 18). The authors find that while forbidden transitions alone do not fully solve the anomaly, combining their results with a proposed increase in the average slope of allowed shape factors can solve both the rate anomaly and the spectral shoulder simultaneously.
+
+## IX. Conclusion
+
+The authors conclude that:
+
+1. Forbidden transitions above 4 MeV are dominant and show strong deviations from allowed approximations.
+2. A correct treatment of these transitions is essential for accurate reactor antineutrino predictions.
+3. The parametrization of forbidden shape factors provides a robust method for uncertainty estimation.
+4. The combination of forbidden transition effects and slope changes in allowed transitions offers a potential solution to the reactor anomaly and spectral bump.
+
+---
+
+## References
+
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+23. H. Behrens and W. Bühring, *Electron radial wave functions and nuclear beta-decay* (Clarendon Press, Oxford, 1982).
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+26. H. Behrens and J. Jänecke, *Landolt-Börnstein Tables, Gruppe I, Band 4* (Springer, 1969).
+27. M. Morita and R. S. Morita, *Physical Review* **109**, 2048 (1958).
+28. H. F. Schopper, *Weak Interactions and Nuclear Beta Decay* (North-Holland Publishing Company, 1966).
+29. T. Kotani, *Physical Review* **114**, 795 (1959).
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+40. M. Kellett, O. Bersillon, and R. Mills, *The JEFF-3.1-3.1.1 RadioActive Decay Data and Fission Yields Sub-Libraries* (JEFF Report 20, 2009).
+41. M. Fallot and A. Sonzogni, *Private Communication*.
+42. M. Herman and A. Trkov, *Brookhaven National Laboratory, Tech. Rep.* (2010).
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+46. L. Hayen and N. Severijns, *Computer Physics Communications* **240**, 152 (2019).
+47. N. Haag et al., *arXiv:1405.3554* (2014).
+48. B. A. Brown and W. D. M. Rae, *Nuclear Data Sheets* **120**, 115 (2014).
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+50. R. Machleidt, *Physical Review C* **63**, 024001 (2001).
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+53. J. Suhonen, *Frontiers in Physics* **5**, 55 (2017).
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+56. M. Haaranen, J. Kotila, and J. Suhonen, *Physical Review C* **95**, 024327 (2017).
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+58. J. P. Davidson, *Collective Models of the Nucleus* (Academic Press, New York and London, 1968).
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+60. A. A. Sonzogni, E. A. McCutchan, and A. C. Hayes, *Physical Review Letters* **119**, 112501 (2017).
+61. D.-L. Fang, *Private Communication*.
+62. T. Yoshida, T. Tachibana, S. Okumura, and S. Chiba, *Physical Review C* **98**, 41303 (2018).
+63. D. W. Scott, *Multivariate Density Estimation: Theory, Practice, and Visualization* (John Wiley & Sons, 1992).
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@@ -0,0 +1,475 @@
+# Summary: High Precision Measurement of the ⁹⁹Tc β Spectrum
+
+## Bibliographic Information
+
+| Field | Value |
+|---|---|
+| **Title** | High precision measurement of the ⁹⁹Tc β spectrum |
+| **Journal** | Physical Review C **110**, 055503 (2024) |
+| **DOI** | 10.1103/PhysRevC.110.055503 |
+| **Received** | 5 October 2023 → Revised 16 August 2024 → Accepted 22 October 2024 → Published 20 November 2024 |
+| **Authors & Affiliations** | M. Paulsen (PTB Berlin), P. C.-O. Ranitzsch, K. Kossert, L. Bockhorn, O. Nähle (PTB Braunschweig); M. Loidl, M. Rodrigues, X. Mougeot†, A. Singh (LNHB/LNE-CEA, Paris-Saclay); J. Beyer, C. Enss (KIP Heidelberg); S. Leblond (LNHB); S. Kempf, M. Wegner, L. Bockhorn‡ (KIT) |
+| **HAL ID** | cea-04796921 |
+
+---
+
+## 1. Objective & Motivation
+
+The paper reports highly precise measurements of the **⁹⁹Tc beta decay spectrum** using **Metallic Magnetic Calorimeters (MMCs)** in two independent laboratories (LNHB, France and PTB, Germany). The goals are:
+
+- Measure the β spectrum shape down to energies < 1 keV — far below previous experiments (>50 keV threshold)
+- Extract the Q-value (maximum β energy) with unprecedented precision
+- Determine the effective axial-vector coupling constant *g*ₐ via the **spectrum-shape method**
+- Derive the average β energy, log *f*, and log *ft* for nuclear reactor modeling
+
+⁹⁹Tc is a second forbidden nonunique transition (β⁻, 99.99855%) decaying to the ⁹⁹Ru ground state. Its spectrum shape was theoretically predicted (Kostensalo & Suhonen, PR C **96**, 024317 (2017)) to be highly sensitive to *g*ₐᵉᶠᶠ, making it an ideal test case.
+
+---
+
+## 2. Decay Scheme (Fig. 1)
+
+⁹⁹Tc → ⁹⁹Ru + e⁻ + ν̄ₑ with:
+- Pure β emission [β⁻, 99.99855(30)%] to the ⁹⁹Ru ground state (reference [11])
+- Q-value previously accepted as 297.5(9) keV (AME2020, reference [17])
+
+---
+
+## 3. Experimental Methods & Setups
+
+### 3.1 Metallic Magnetic Calorimeters (MMC) — Principle
+
+MMCs are cryogenic microcalorimeters where:
+- A metallic absorber is in thermal contact with a paramagnetic sensor (Ag:Er, 300 ppm)
+- Energy deposition → temperature rise ΔT → magnetization change ΔM via:
+
+**Eq. (1):**  
+Δ*M* = (∂*M*/∂*T*) · Δ*T* = (*E* / ∂*T*) · (∂*M*/∂*T*) · (Δ*T*/*C*_tot)
+
+The magnetization change is picked up by a superconducting coil and measured with a SQUID.
+
+### 3.2 Setup Comparison: LNHB vs PTB
+
+| Parameter | **LNHB** [10,20,21] | **PTB** [28,29] |
+|---|---|---|
+| Cryostat | ¾³He dilution insert in ⁴He(l) bath | ¾³He dilution refrigerator with two-stage pulse tube precooling |
+| MMC chip | MetroBeta V1-M | MetroBeta V2-M |
+| SQUID chip | Supracon VC1A | PTB X1 |
+| Input inductance | 4.5 nH | 2 nH |
+| Absorber (Au, heat capacity) | 350 pJ/K at 20 mK | 112 pJ/K at 20 mK |
+| Sample prep | **Electrodeposited** on Au foil → diffusion welded into absorber | **Drop deposited** from NH₄⁹⁹TcO₄ solution onto Au substrate |
+| Calibration source | ¹³³Ba (~50 kBq) | ⁵⁷Co (~37 kBq) |
+| Analysis code | Optimal filtering in **MATLAB** | Optimal filtering in **PYTHON** |
+
+#### LNHB Details:
+- Operating temperature: *T* = 12 mK
+- Data acquisition: continuous stream over 13.7 days at 100 kS/s
+- Final spectrum: **7,264,451 events**, energy resolution ΔE ≈ 100 eV (constant up to 384 keV)
+
+#### PTB Details:
+- Operating temperature: *T* = 14.5 mK  
+- Data acquisition: ~20 days at 200 kS/s
+- Final spectrum: **5,326,682 counts** (after cuts)
+- Second pixel used for background measurement
+
+### 3.3 Silicon Detector Setup — PIPS at LNHB
+
+Independent confirmation using two passivated implanted planar silicon (PIPS) detectors in a quasi-4π configuration:
+- Source placed between facing detectors → >98% solid angle coverage
+- Ultrahigh vacuum, cooled to 100 K with liquid nitrogen
+- Activity ~800 Bq, dead-time <0.5%
+- Energy resolution: **9 keV at 65.52 keV** (two orders of magnitude worse than MMCs)
+- Detection threshold: ~10 keV (above MMC's sub-keV threshold)
+
+---
+
+## 4. Calibration & Linearity Correction
+
+### 4.1 Energy Calibration Lines (Table III — key entries)
+
+**LNHB (¹³³Ba source):**
+| Origin | Tabulated (keV) | Measured (keV) | ΔE (keV) | Type |
+|---|---|---|---|---|
+| ¹³³Ba Cs Kα₂ | 30.63 | 30.61 | −0.02 | X-ray |
+| ¹³³Ba Cs Kα₁ | 30.97 | 30.97 | 0.00 | X-ray |
+| Pb fluorescence (Kα₂) | 72.81 | 72.82 | +0.02 | X-ray |
+| Pb fluorescence (Kα₁) | 74.97 | 74.98 | +0.01 | X-ray |
+| Escape lines (356 keV γ − Au Kα₂) | 289.02 | 289.08 | +0.06 | Escape |
+| ¹³³Ba γ (direct) | 383.85 | 383.82 | −0.03 | Gamma |
+
+**PTB (⁵⁷Co source):**
+| Origin | Tabulated (keV) | Measured (keV) | ΔE (keV) | Type |
+|---|---|---|---|---|
+| ⁵⁷Co γ | 14.42 | 14.44 | −0.02 | Gamma |
+| Au Kα₁ fluorescence | 68.80 | 68.95 | −0.15 | X-ray |
+| Pb Kβ₁ fluorescence | 84.94 | 85.09 | −0.15 | X-ray |
+| Escape lines (122 keV γ) | ~53–55 | ±0.17 shift | — | Escape |
+
+### 4.2 Polynomial Calibration Coefficients (Table II)
+
+The energy scale was corrected for nonlinearities using a second-order polynomial:
+
+**Eq. (2):**  
+*E*_lit = *f_fit*(aᵢ) := k₁ · aᵢ + k₂ · aᵢ²
+
+| | **LNHB (Set 3)** | **PTB (Set 2)** |
+|---|---|---|
+| Linear coefficient k₁ (keV) | 355.822 | 121.355 |
+| Quadratic coefficient k₂ (keV) | 0.183 | 0.706 |
+| Linearity offset kγ₁ − 1 | 0.1% | 0.6% |
+
+The PTB nonlinearity is larger due to different detector design and thermalization.
+
+---
+
+## 5. Spectrum Corrections
+
+### 5.1 Background Subtraction
+- **LNHB approach:** Constant background + step functions below calibration peaks (since several peaks are near/above the β endpoint)
+- **PTB approach:** Dedicated second pixel provides direct background spectrum, convolved with Gaussian and scaled before subtraction
+
+### 5.2 Energy Loss Unfolding (Eq. 3–4)
+
+β electrons lose energy via Au x-ray fluorescence and bremsstrahlung. Correction uses discrete unfolding:
+
+**Eq. (3):** *h*_N^meas = R_N×N · *h*_N^true
+
+where the response matrix R is obtained from Monte Carlo simulations using **EGSNRC/egs_phd** [42].
+
+**Eq. (4):** *h*_N^algo ≈ (*R_sim*)⁻¹ · *h*_N^meas
+
+The unfolding shifted the maximum energy by approximately **+200 eV**. Energy losses were very small (<1%).
+
+---
+
+## 6. Cross-Validation Results (Table IV)
+
+| Set | Measurement | Analysis | Counts | FWHM | ETH |
+|---|---|---|---|---|---|
+| 1 | PTB | LNHB code | 3.67×10⁶ | 63 eV @ 136 keV | 345 eV |
+| 2 | PTB | PTB code | 5.33×10⁶ | 72 eV @ 136 keV | 750 eV |
+| 3 | LNHB | LNHB code | 7.26×10⁶ | 150 eV @ 303 keV | 1120 eV |
+| 4 | LNHB | PTB code | 5.66×10⁶ | 108 eV @ 303 keV | 1250 eV |
+
+All four spectra show excellent agreement — confirming the results are not dependent on analysis software or detector design.
+
+---
+
+## 7. Key Results: Q-Value & Coupling Constants
+
+### 7.1 Maximum β Energy / Q-value (Table V)
+
+The spectrum-shape method fit gives *E_max* for all four data sets:
+
+| Set | Measurement | Analysis | *E_max* (keV) | λ |
+|---|---|---|---|---|
+| 1 | PTB | LNHB | **295.809** | 0.615 |
+| 2 | PTB | PTB | **295.786** | 0.652 |
+| 3 | LNHB | LNHB | **295.845** | 0.651 |
+| 4 | LNHB | PTB | **295.862** | 0.659 |
+
+**Final weighted mean:**  
+***E_max* = *Q_β* = 295.82(16) keV** ← **Eq. (9)**
+
+This is **five times more precise** than the previously recommended value of 297.5(9) keV, and shifted by −1.8 keV (~2σ).
+
+### 7.2 Uncertainty Budget — Q-value (Table VI)
+
+| Component | LNHB u(eV) | PTB u(eV) |
+|---|---|---|
+| Energy calibration | 45 eV | 50 eV |
+| Background | **141 eV** (LNHB) / 69 eV (PTB) — *dominant* | |
+| Fit method | 93 eV | 89 eV |
+| Theoretical model | 75 eV | 75 eV |
+| Spectrum unfolding | 57 eV | 57 eV |
+
+### 7.3 Effective Axial-Vector Coupling Constant (Table VIII)
+
+The spectrum-shape method fits *g*ₐᵉᶠᶠ by comparing measured spectrum to shell-model predictions:
+
+**Best fit value:**  
+***g*ₐᵉᶠᶠ = 1.526(92)** — **Eq. (15)**, unquenched!
+
+After renormalizing half-life agreement:  
+***g*_V^eff* = 0.376(5) and *g*_A^eff* = 0.574(36)** ← **Eq. (16)**
+
+These quenched values are consistent with those for **first forbidden nonunique transitions**, solving a long-standing inconsistency noted by Suhonen [56].
+
+### 7.4 Derived Quantities — Eqs. (17)–(19)
+
+| Quantity | Value |
+|---|---|
+| Average β energy, ⟨*E_β*⟩ | **98.51(23) keV** ← Eq. (17) |
+| log *f* | **−0.476 60(22)** ← Eq. (18) |
+| log *ft* | **12.3478(23)** ← Eq. (19) |
+
+For comparison, the experimental shape factor from literature [15] gives: ⟨*E_β*⟩ = 95.91(5) keV — showing that old parametrizations are now obsolete.
+
+---
+
+## 8. Nuclear Shell Model Calculations
+
+### 8.1 Theoretical Framework (Section III.B)
+
+The β spectrum is described by the Behrens–Bühring formalism:
+
+**Eq. (5):**  
+*N*(*W*)d*W* = (*G*_β² / 2π³) · *F*(*Z*,*W*)·p*W*·(*W₀*−*W*)² · X(*W*)·C(*W*)·r(*Z*,*W*)
+
+Where:
+- *F*(*Z*,*W*) = Fermi function
+- *X*(*W*) = atomic screening and exchange corrections  
+- C(*W*) = shape factor (second forbidden nonunique)
+- r(*Z*,*W*) = atomic overlap correction, with B̈″ = 0.211(11) keV
+
+**Eq. (6):** r(*Z*,*W*) = 1 − [1/(*W₀*−*W*)] · ∂²B(G)/∂Z²
+
+### 8.2 NUSHELLX Calculations
+
+Three effective interactions were tested:
+1. **Gloeckner** [66] — GL valence space (p₁/₂, g₉/₂ for protons; s₁/₂, d₅/₂ for neutrons) → single transition dominates: *n*(2d₅/₂) → *p*(1g₉/₂)
+2. **Mach** [67] — wider GLEKPN valence space with blocked orbitals → small admixture from *n*(1g₇/₂) → *p*(1g₉/2)
+3. **jj45pnb** [68] — different sign for dominant OBTDs
+
+### 8.3 One-Body Transition Densities (Table VII)
+
+| Transition | OBTD | ΔEC(3) MeV | ΔEC(4)₁,₂ MeV | ΔEC(4)₂,₁ MeV |
+|---|---|---|---|---|
+| 1g₇/₂ → 1g₉/₂ | 0.00994 | 11.696 | 12.051 | 12.045 |
+| **2d₅/₂ → 1g₉/₂** | **0.47752** (dominant) | 10.560 | 10.536 | 10.537 |
+
+### 8.4 Coulomb Displacement Energy (Eq. 11–14)
+
+The relationship between relativistic and nonrelativistic vector matrix elements:
+
+**Eq. (11):** V²₂₁ᶠ ≃ −(*R*/√10)[*W₀* − (*mₙ*−*mₚ*) + ΔEC] · V²₂₀ᶠ
+
+Different estimates for ΔEC were tested:
+- **ΔEC⁽¹⁾** = 13.476 MeV (uniformly charged sphere) — least realistic
+- **ΔEC⁽²⁾** = 12.814 MeV (parent+daughter radii)
+- **ΔEC⁽³⁾** = 10.5 MeV (average Coulomb potential, leading order) ← used as reference
+- **ΔEC⁽⁴⁾** = full lepton current treatment — most accurate
+
+---
+
+## 9. Figures Summary
+
+| Figure | Description | Key Finding |
+|---|---|---|
+| **Fig. 1** | Decay scheme of ⁹⁹Tc | Pure β to ⁹⁹Ru ground state |
+| **Fig. 2** | Schematic overview of workflow | Data → evaluation → theory → extraction pipeline |
+| **Fig. 3** | LNHB MMC spectrum with calibration peaks (¹³³Ba) | Spectrum down to <1 keV, inset shows endpoint region |
+| **Fig. 4** | PTB MMC spectrum with calibration peaks (⁵⁷Co) | Similar quality, different source prep & analysis code |
+| **Fig. 5** | All four corrected spectra compared | Excellent agreement across all sets in linear and log scale |
+| **Fig. 6** | Normalized residual plots between analyses | Residuals <5% below 1.5 keV — cross-analysis validates methods |
+| **Fig. 7** | MMC + PIPS comparison (above 25 keV) | Three independent measurements agree perfectly; old literature spectrum diverges below 100 keV |
+| **Fig. 8** | Theoretical spectra vs measurement (varying ΔEC & CVC) | Spectrum mostly sensitive to low energy; CVC has minor effect |
+| **Fig. 9** | Effect of *g*ₐᵉᶠᶠ on spectrum shape | Strong sensitivity — *g*ₐᵉᶠᶠ between 1.4–1.6 gives good agreement |
+| **Fig. 10** | Best-fit theoretical curve (*g*ₐᵉᶠᶠ = 1.526) vs data | Excellent agreement down to 6 keV; residuals are Gaussian centered at zero |
+
+---
+
+## 10. Analysis Techniques & Methods Summary
+
+| Technique | Purpose | Implementation |
+|---|---|---|
+| **Optimal filtering** | Pulse height estimation from raw ADC data | MATLAB (LNHB) / PYTHON (PTB) |
+| **Second-order polynomial calibration** | Energy linearity correction across full range | Orthogonal distance regression (scipy.odr) |
+| **Background subtraction** | Remove ¹³³Ba/⁵⁷Co + environmental background | Constant + step functions (LNHB); direct pixel measurement (PTB) |
+| **Monte Carlo unfolding** | Correct for energy losses in absorber | EGSNRC response matrix inversion |
+| **Spectrum-shape fitting** | Extract *Q_β*, *g*ₐᵉᶠᶠ by comparing to theory | χ² minimization over theoretical curves |
+| **Nuclear shell model (NUSHELLX)** | Calculate OBTDs and shape factor C(*W*) | Three different effective interactions tested |
+
+---
+
+## 11. Overall Article Logic / Structure
+
+```
+Introduction → Motivation: why ⁹⁹Tc β spectrum matters
+   ↓
+Sec II — Experimental Study (3 measurements, independent setups)
+   ├─ A. MMC at LNHB & PTB + cross-analysis
+   └─ B. PIPS detector confirmation (>25 keV)
+   ↓
+Sec III — Combined Analysis
+   ├─ A. Q-value from Kurie-like fit → 295.82(16) keV
+   └─ B. Spectrum-shape method: shell model + gA extraction
+       ├─ Theoretical modeling (Behrens-Bühring formalism)
+       ├─ Nuclear structure (NUSHELLX, OBTDs)
+       ├─ Coulomb displacement energy estimates
+       └─ Effective coupling constants → quenched values
+   ↓
+Sec IV — Conclusion: Qβ confirmed, gAᵉᶠᶠ solved inconsistency, 
+         recommendations for Penning trap confirmation
+```
+
+---
+
+## 12. Conclusions & Implications
+
+1. **Q_β = 295.82(16) keV** — 5× more precise than previous value; −1.8 keV shift calls for Penning trap verification
+2. ***g*_A^eff* = 0.574(36)** (quenched) — resolves inconsistency between first and higher forbidden transitions
+3. **⟨*E_β*⟩ = 98.51(23) keV** — directly applicable to nuclear reactor decay heat calculations
+4. MMCs are established as the premier technique for low-energy β spectrometry (sub-keV threshold, ~100 eV resolution)
+5. Literature parametrizations of the ⁹⁹Tc spectrum must now be considered obsolete
+
+---
+
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+
+---
+
+## Extraction Quality Notes
+
+| Content Type | Extraction Quality | Notes |
+|---|---|---|
+| Full text | ✅ Excellent | All ~1180 lines extracted cleanly via pdftotext -layout |
+| Tables | ⚠️ Partially | pdfplumber returned mostly empty; full content recovered from layout text above each table |
+| Formulae | ✅ Good | LaTeX-style notation preserved (some Unicode artifacts like `99` superscripts) |
+| Figures | 📋 Described | 3 images extracted (page1: logo + decay scheme; page2: small icon); cannot render but captions and content described from text |
+| Bibliography | ✅ Complete | All 73 references extracted in full with DOIs/journal info |
@@ -0,0 +1,176 @@
+Here is the updated summary with the inline citations integrated as requested.
+
+# Summary: The measurement of the $^{99}$Tc $\beta$-decay spectrum and its implications for the effective value of weak axial coupling
+
+## 1. Introduction and Motivation
+
+The study of $\beta$-spectral shapes is a critical method for determining the effective value of the weak axial coupling constant, $g_A$. In forbidden non-unique $\beta^-$ transitions [5], the spectral shape is directly sensitive to the ratio $g_A/g_V$.
+
+* **Standard Hypothesis:** The Conserved Vector Current (CVC) hypothesis sets the vector coupling constant $g_V = 1.0$ [8].
+* **Quenching:** The Partially Conserved Axial-Vector Current (PCAC) hypothesis suggests that in finite nuclei, $g_A$ is "quenched" (reduced) to a value of approximately $1.0$, compared to the bare-nucleon value of $1.27$ resulting from the experimental analysis of the decay of an isolated neutron [8]. The quenching of the effective value of $g_A$ has been studied extensively in [2, 8, 9], and in particular its effects on the neutrinoless double beta decay have been addressed in [1, 3].
+* **Contradiction:** While previous studies on the fourth-forbidden non-unique $\beta^-$-decays of $^{113}$Cd [10–13] and $^{115}$In [14–16] supported quenching ($g_A/g_V \approx 1.0$), a recent measurement by Paulsen et al. [17] on the second-forbidden non-unique transition in $^{99}$Tc suggested an *enhanced* ratio of $g_A/g_V = 1.526(92)$.
+
+This paper aims to resolve this contradiction by performing a high-precision measurement of the $^{99}$Tc $\beta$-decay spectrum using Metallic Magnetic Calorimeters (MMCs) and comparing it with state-of-the-art nuclear theory.
+
+## 2. Theoretical Framework
+
+The half-life ($t_{1/2}$) and shape of the $\beta$-spectrum are governed by the integrated shape function $\tilde{C}$.
+
+**Equation 1:**
+$$ t_{1/2} = \frac{\kappa}{\tilde{C}} $$
+*(where $\kappa$ is a constant [18, 19] and $\tilde{C}$ incorporates phase-space factors and NMEs arising in the next-to-leading-order expansion, as discussed in detail in Refs. [6, 7].)*
+
+The shape function $C(w_e)$ depends on the weak axial and vector couplings:
+
+**Equation 2:**
+$$ C(w_e) = g_V^2 C_V(w_e) + g_A^2 C_A(w_e) + g_V g_A C_{VA}(w_e) $$
+*(where $w_e$ is the total energy of the emitted electron. The transition matrix element [6].)*
+
+**Methodology (ESSM):**
+The authors employ the **Enhanced Spectrum-Shape Method (ESSM)** [19]. This approach simultaneously fits the measured $\beta$-spectrum and the experimental partial half-life to extract $g_A^{eff}$. This allows for the adjustment of the small relativistic vector nuclear matrix element (sNME) [5] to reproduce the half-life, rather than fixing it to the "ideal" CVC value.
+
+The sNME gathers contributions outside the valence major shell, making its calculation particularly hard for nuclear-theory frameworks used, e.g., in the present work and also in the work of Paulsen et al. [17]. Despite its smallness, the sNME can influence the $\beta$-spectral shapes and half-lives quite strongly [11, 12, 19, 21].
+
+A realistic value of the sNME can be estimated from its CVC value [5], first used for spectral-shape calculations in Ref. [18] and later by Paulsen et al. [17]. To circumvent the use of the CVC value, the experimental works [11, 13, 15] and theoretical works [12, 16] used sNME as a fitting parameter, together with $g_A^{eff}$.
+
+In this Letter, we present new theoretical calculations of the $\beta$ spectrum of $^{99}$Tc... In references [21, 23] the related two sets of theoretical $\beta$-spectra of $^{99}$Tc were introduced for two NSM Hamiltonians, namely $jj45pnb$ [24] and $glekpn$ [25].
+
+## 3. Experimental Setup
+
+The experiment was conducted at Lawrence Livermore National Laboratory (LLNL).
+
+* **Detector:** A Metallic Magnetic Calorimeter (MMC) fabricated at KRISS [26, 27].
+  * **Sensor:** Paramagnetic ions (420 ppm of Ag:$^{168}$Er) embedded in a metallic host.
+  * **Absorber:** A 4$\pi$ solid-angle gold absorber containing the $^{99}$Tc source.
+  * **Temperature:** Operated below 100 mK (specifically 15 mK during data taking).
+* **Source Preparation:** $^{99}$Tc solution was dropped onto a gold foil, dried, folded, and rolled (kneading method [28, 29]) to ensure uniform distribution. The final sample mass was 1.75 mg.
+* **Readout:** A Magnicon SQUID (Model X1) and an additional array SQUID (Model X16).
+* **Data Acquisition:** Continuous mode, 200 kS/s sampling rate using a Stanford Research Systems SRS 560 amplifier [30].
+* **Threshold:** A low-energy threshold of 10 keV was applied.
+
+**Figure 1: Detector Setup**
+*(Description of visual content)*
+The image shows the experimental apparatus. Key components labeled include:
+
+* **Magnicon SQUID X1:** The signal readout device.
+* **KRISS MMC:** The calorimeter sensor.
+* **$^{99}$Tc source:** The radioactive sample placed on the sensor.
+* **Support washer & Kapton tape:** Used for thermal insulation and alignment.
+
+## 4. Results and Analysis
+
+### 4.1 Data Selection
+
+* **Background Dataset:** 130,011 events recorded.
+* **Signal Dataset:** After Pulse Shape Discrimination (PSD) cuts, **77,487 events** corresponding to $^{99}$Tc remained for analysis.
+
+### 4.2 Energy Calibration
+
+Calibration was performed using an external $^{133}$Ba source and internal X-rays from a $^{148}$Gd source (used for internal calibration checks).
+
+**Table I: Measured X-ray and gamma-ray lines**
+
+| Origin | Radiation type | Tabulated energy [keV] | Measured energy [keV] | FWHM resolution [keV] | $\Delta$E [keV] |
+| :--- | :--- | :--- | :--- | :--- | :--- |
+| **Gd fluorescence** | X-ray K$_{\alpha2}$ | 42.3097(13) | 42.30(10) | 0.9(4) | 0.01(10) |
+| | X-ray K$_{\alpha1}$ | 42.9968(12) | 42.9427(31) | 0.68(10) | 0.05(33) |
+| | X-ray K$_{\beta3}$ | 48.5505(21) | 48.29(5)* | 1.05(22) | 0.41(5) |
+| | X-ray K$_{\beta1}$ | 48.6957(21) | | | |
+| **Pt fluorescence** | X-ray K$_{\alpha2}$ | 65.1226(21) | 65.005(22) | 0.9(6) | 0.118(22) |
+| | X-ray K$_{\alpha1}$ | 66.8311(20) | 66.78(5) | 0.65(15) | 0.06(5) |
+| **$^{133}$Ba calibration** | $\gamma$ | 80.9979(11) | 80.881(24) | 0.77(8) | 0.117(25) |
+| | $\gamma$ | 276.3989(12) | 276.12(5) | 0.57(13) | 0.28(5) |
+| | $\gamma$ | 302.8508(5) | 302.764(21) | 0.60(5) | 0.087(21) |
+| | $\gamma$ | 356.0129(7) | 355.926(9) | 0.653(21) | 0.087(9) |
+| | $\gamma$ | 383.8485(12) | 383.69(6) | 0.82(14) | 0.16(6) |
+
+*(Note: Entries labeled "Not distinguished" in the original text refer to peaks with low relative intensities that could not be individually resolved.)*
+
+### 4.3 Spectral Comparison
+
+The measured spectrum was compared against theoretical predictions from three nuclear models:
+
+1. **$jj45pnb$** (NSM Hamiltonian)
+2. **$glekpn$** (NSM Hamiltonian)
+3. **MQPM** (Microscopic Quasiparticle-Phonon Model)
+
+**Figure 2: $^{99}$Tc spectrum with external $^{133}$Ba calibration source**
+*(Description)*
+A plot showing "Counts / 200 keV" vs "Energy (keV)". It displays the raw data (black line) with distinct peaks corresponding to Ba-133 $\gamma$-lines, Au X-rays, Pt X-rays, and Gd X-rays [34].
+
+**Figure 3: Spectrum of calibrated $^{99}$Tc compared with theoretical predictions**
+*(Description)*
+A plot showing "Counts / 1 keV" vs "Energy (keV)" (0-300 keV).
+
+* **Black line:** Experimental Data.
+* **Blue dashed line:** $jj45pnb$ model ($g_A=1.0$).
+* **Green dotted line:** $glekpn$ model ($g_A=1.2$).
+* **Red dash-dot line:** MQPM model ($g_A=1.0$).
+* **Bottom panel:** Residuals ($\Delta/c$) showing the fit quality over the 0-280 keV range.
+
+### 4.4 Quantitative Results (Table II)
+
+The table below summarizes the fit results for the effective axial coupling $g_A^{eff}$ and the goodness of fit ($\chi^2_\nu$ and p-value).
+
+**Table II: Results for the three considered nuclear models**
+
+| Model | $g_A^{eff}$ | sNME [fm$^3$] | $\chi^2_\nu$ | p-value |
+| :--- | :--- | :--- | :--- | :--- |
+| **CloseCVC** | | | | |
+| glekpn | 1.1 | 0.0674 | 1.1057 | 0.12 |
+| jj45pnb | 1.0 | 0.0681 | 1.2436 | 0.004 |
+| MQPM | 1.0 | 0.0651 | 1.6437 | <0.0001 |
+| **FarCVC (Best fit)** | | | | |
+| glekpn | 1.2 | -0.0698 | 1.0664 | 0.22 |
+| jj45pnb | 1.0 | -0.0669 | 1.0966 | 0.14 |
+| MQPM | 1.0 | -0.0683 | 1.0951 | 0.14 |
+
+* **CloseCVC:** Assumes the sNME is fixed to the CVC value. The $jj45pnb$ and MQPM models show poor agreement (low p-values).
+* **FarCVC (Best fit):** Allows sNME to vary. The **$glekpn$ model with $g_A^{eff} = 1.2$** yields the best agreement ($\chi^2_\nu = 1.0664$, p-value = 0.22). The other models with $g_A^{eff} = 1.0$ also show statistically compatible fits (p-values $\approx$ 0.14).
+
+## 5. Conclusion
+
+The high-precision measurement of the $^{99}$Tc $\beta$-decay spectrum supports a **quenched axial coupling** ($g_A^{eff} \approx 1.0 - 1.2$).
+
+* This contradicts the previously reported enhanced value ($g_A/g_V = 1.526$) by Paulsen et al. [17].
+* The discrepancy is attributed to the different treatment of the sNME (small relativistic vector nuclear matrix element) in the theoretical analysis.
+* The results are consistent with quenching patterns observed in heavier nuclei, such as $^{113}$Cd [10–13] and $^{115}$In [14–16].
+
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